WASHINGTON — The White House and Senate Minority Leader Mitch McConnell, R-Ky., agreed on the outlines of a bipartisan deal Sunday to raise the federal debt limit as the clock ticked toward a Tuesday deadline to avert a possible government default.

Senate Minority Leader Harry Reid, D-Nev., issued a statement expressing his support for the emerging framework, which would effectively avert another grueling battle over the debt limit until after the 2012 presidential elections. But the proposal would make no explicit provision to raise taxes, which Democrats had sought.

Tea-party-aligned Senate Republicans also appeared to be falling into line. Although they would not support a debt-limit increase, several said they would drop their threat to block Senate action, which could come as soon as Monday.

While the legislative path appeared to be clearing in the Senate, however, the House remained a forbidding thicket. House Speaker John Boehner, R-Ohio, was holed up in his Capitol office and remained largely silent Sunday. Aides in both parties said Boehner had yet to sign off on the deal, which would require him to sell significant cuts in military spending to defense hawks in his party — including many House veterans Boehner will need to push the compromise to final passage.

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A House Republican leadership aide who was not authorized to speak publicly about the talks said that discussions were moving in the right direction but that “serious issues” remained. The aide added Boehner would make no decisions until his fractious caucus had a chance to weigh in. Boehner scheduled a conference call with GOP lawmakers for Sunday evening.

Asked on “Fox News Sunday” if the deal could win approval from the conservative House GOP caucus, the party’s chief vote counter, Majority Whip Kevin McCarthy, R-Calif., was noncommittal.

“I think the American people want a balanced budget,” McCarthy said, referring to the GOP’s push for a constitutional amendment requirement a balanced budget. “That’s where we stand, and that’s what we’ll fight for.”

The plan McConnell negotiated with the White House requires only that each chamber hold votes on a balanced-budget amendment, a weaker version than the legislation House Republicans backed last week.

In an appearance on CNN’s “State of the Union,” McConnell sketched the outlines of the developing deal, which would raise the debt limit by as much as $2.4 trillion in two stages. The first increase would total $900 billion, aides said, with the Treasury gaining access to $400 billion in additional borrowing authority immediately upon congressional passage of the deal. The other $500 billion would come later this fall, unless two-thirds of the members of both chambers of Congress objected.

The second increase would hike the debt limit by at least $1.2 trillion, also subject to a similar resolution of congressional disapproval. This provision would place the entire burden for a debt-limit increase on the White House and Democrats, since Republicans would be expected to object but not in numbers large enough to stop the ceiling from being raised.

The second hike could be larger than $1.2 trillion, however, if a new bipartisan committee tasked with drafting a broader debt-reduction plan comes up with an even bigger package of savings.

The framework builds on similar measures introduced by Boehner and Reid. The Boehner bill passed the House but was shot down in the Senate. The Senate rejected Reid’s bill Sunday on a largely party-line vote, 50 to 49.

For days, the chief obstacle to a deal was the design of a mechanism to force the committee to act – or to make sure spending cuts were adopted if the committee failed. In the end, negotiators settled on a trigger that would force automatic across-the-board cuts of $1.2 trillion to agency budgets, split half and half between domestic programs and defense.

Programs for the poor, including Medicaid and Social Security, would be exempted. But Medicare payments to providers could be hit.

The framework represents a limited victory for President Barack Obama, meeting his demand to avoid another debt-limit fight in the heat of the 2012 campaign. However, Obama failed to secure his other top priorities, including fresh measures to revive the flagging recovery and an agreement to end tax breaks for corporations and the wealthy.

Republicans, by contrast, won an upfront agreement to cut nearly $1 trillion from agency budgets over the next decade and the prospect of deeper cuts to come. Democrats also agreed to stage a vote on a balanced-budget amendment, a rallying point for conservatives. And while the trigger would hit defense, a sensitive matter for Republicans, it would not raise taxes.

“What I tell people at home is that instead of adding $10 trillion to the debt, we’re going to add $7 trillion” through 2021, McConnell said on CBS’s “Face the Nation.

Liberals were not enthusiastic about the package, arguing that it entailed surrender on all their top priorities. House Minority Leader Nancy Pelosi, D-Calif., withheld comment, saying she would confer with House Democrats on Monday.

Rep. Raul Grijalva, D-Ariz., who heads the 74-member Congressional Progressive Caucus, was blunt about his opposition.

“This deal trades peoples’ livelihoods for the votes of a few unappeasable right-wing radicals, and I will not support it,” he said.

Senate Majority Whip Richard Durbin, D-Ill., said the decision to take “revenues off the table I think is a serious mistake.”

But David Plouffe, a senior White House political adviser, defended the package on ABC’s “This Week,” saying, “The president has been clear he is willing to do some tough things.”

“Our focus now is on solving this,” Plouffe said. “There is no off-ramp here. The only option is for Congress to raise the debt ceiling.”

A bigger stumbling block may be Republicans on the House Armed Services Committee and the Appropriations subcommittee on defense. The two chairmen of those panels, Reps. Buck McKeon, R-Calif., and Bill Young, R-Fla., held a joint news conference Saturday, vowing to oppose any long-term reduction in defense spending that exceeded $439 billion over the next 10 years. That is the number the y approved during recent deliberations over next year’s budget.

Veterans such as McKeon and Young are the base of Boehner’s support within the House Republican conference. Boehner would need to win their votes to pass the debt-limit package, even if Pelosi agrees to provide a significant bloc of Democrats.